A general paradigm for lifting binary morphological algorithms to fuzzy algorithms is employed to construct fuzzy versions of classical binary morphological operations. The lifting procedure is based upon an epistemological interpretation of both image and filter fuzzification. Algorithms are designed via the paradigm for various fuzzifications and their performances are analyzed to provide insight into the kind of liftings that produce suitable results. Algorithms are discussed for three image processing tasks: shape detection, edge detection, and clutter removal. Detailed analyses are given for the effect of noise and its mitigation owing to fuzzy approaches. It is demonstrated how the fuzzy hit-or-miss transform can be used in conjunction with a decision procedure to achieve word recognition.

In all man-machine systems with image processing functions, an important unsolved problem arises in the treatment of uncertain and incomplete image information. Several frameworks have been suggested for handling uncertain image information including; expert systems, fuzzification, likelihood estimation, and neural networks. In this paper we review those methods. We also present a new method for handling uncertainties by unifying the representations of gray-values and uncertainty into one framework in a way that parallels fuzzy logic. This new framework is based on the application of the extended fuzzy pointed set and an associated algebra to handle uncertain information. We further show how this framework can be used in image processing and artificial intelligence.

Intrinsic fuzzification of mathematical morphology is grounded on an axiomatic characterization of subset fuzzification. The result is an axiomatic formulation of fuzzy Minkowski algebra. Part of the Minkowski algebra results solely from the axioms themselves and part results from a specific postulated form of a subsethood indicator function. There exists an infinite number of fuzzy morphologies satisfying the axioms; in particular, there are uncountably many indicators satisfying the postulated form. This paper develops fuzzy Minkowski algebra, with special emphasis on fitting characterizations of fuzzy erosion and opening, examines key properties of the indicator function, and provides fuzzy extensions of the basic binary Matheron representations for openings and increasing, translation-invariant operators.

Two morphological algorithms that attempt to recognize a black and white object directly in its discrete domain are presented. The first algorithm is based on covariance functions, while the second is based on a variant of size distribution functions. In both these algorithms, the scale correction has been automated. Also presented is a complete geometric and algebraic characterization of objects that are identical with respect to the proposed methodologies, and it is shown that the induced equivalent classes over binary images contain objects that are structurally very similar. This has been accomplished by introducing the notions of a strongly attached pixel, discrete structure of an image, and a structure preserving operation. An outcome of the analysis is the insight into the relationship between the discrete structure of an image and the induced equivalence classes.>

A collection of all plausible solutions to a given complex problem is considered, and it is assumed that these solutions are separately ranked by the degree to which they satisfy each member of an a priori chosen criteria set. Oftentimes these rankings can be accurately modeled as partial orders. Given such a collection of partially ordered sets (posets), one may wish to define a class of operators whose members may be used to aggregate this collection of posets into a single poset. Selection of a solution to the given problem can then be performed on this poset directly. The authors propose a framework that allows the construction of a variety of aggregation operators so as to bring forth their algebraic structure.>